The mobile-payments company Square, founded by Jack Dorsey, is finalizing a deal to expand a partnership with the city to get its credit-card reader into more New York City taxis, a report Monday claims.

The soon-to-be-inked deal follows a visit by Dorsey to the Big Apple last week, where the executive chatted with Mayor Michael Bloomberg and the Taxi and Limousine Commission, according to the report, from research firm PrivCo, which posted a photo of Bloomberg and Dorsey taking a cab ride.

“Square will be rolled out in phases: all new cabs will come standard now with an iPad and Square as the payment method, some 1,000 newer cabs will be retrofitted over the course of the next six months with iPads equipped with Square. Older cabs nearing the end of useful life [would be] exempted,” according to Sam Hamadeh, CEO of PrivCo.

However, Square yesterday denied any deal was afoot.

“We have a positive, productive relationship with the TLC,” Square said in an e-mailed statement. “While the Mayor and Jack did have a very enjoyable ride in a Square-equipped cab, PrivCo’s other statements have no basis in fact.”

City officials also denied any such deal is in the works.

An expanded city partnership would be big for Square, potentially adding $12 million a year in revenue, he said.

Square’s pilot program with the city’s taxis, announced in March, covered about 30 taxis. The new deal would cover about 1,200 taxis next year, according to PrivCo.

There are more than 13,000 yellow cabs in the city.

Square takes a 2.75 percent transaction fee for each credit-card swipe. Traditional credit-card transactions come with an up to 4 percent fee.